We are asked to find the values of the variables a, b, and c. Let's do this by considering one variable at a time.
Variable a
We can start with analyzing the given diagram. For the purposes of the solution, let's name some of the points.
As we can see, angle ∠ a intercepts the arc AB. To find the angle measure, we will use the Inscribed Angle Theorem.
Inscribed Angle Theorem
The measure of an inscribed angle is half the measure of its intercepted arc.
According to this theorem, the measure of ∠ a is half the measure of AB.
m∠ a=1/2mAB
From the diagram, we know that mAB= 52^(∘). Let's substitute this value into the equation and calculate the measure of ∠ a.
m∠ a=1/2( 52^(∘))=26^(∘)
Variable b
Now, we will try find the value of b. Let's examine the diagram!
We can see that ∠ b is an angle formed by a tangent to the point C and a chord AC. We can use Theorem 12-12, which states the following.
Theorem 12-12
The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc.
Angle ∠ b intercepts arc AC. If we find the arc's measure, we will be able to calculate m∠ b. Since arc BAC is created by the diameter BC, it measures 180^(∘).
It contains the arcs BA and AC. By the Arc Addition Postulate the following equation is true.
mBAC=mBA+mAC
Let's substitute mBAC with 180^(∘) and mBA with 52^(∘) and solve the equation for mAC.
Finally, by applying Theorem 12-12, we conclude that the measure of ∠ b is half the measure of AC. Let's divide 128 by 2 and calculate the value of b.
b=128/2=64
Variable c
We will calculate the value of variable c. Again, let's start with analyzing the diagram.
As we can see, angle ∠ c intercepts arc BD, which measures 84^(∘). By the stated above Inscribed Angle Theorem, the measure of the angle is half the measure of its intercepted arc. Dividing 84 by 2, we can calculate m∠ c.
m∠ c=1/2( 84^(∘))=42^(∘)
We conclude that the value of c is 42.
